For the control system with a piezo actuator in astrophysical research the condition for the existence of self-oscillations is determined. Frequency method for determination self-oscillations in control systems is applied. By using the harmonious linearization of hysteresis and Nyquist stability criterion the condition of the existence of self-oscillations is obtained.

Keywords: frequency method, control system, piezoactuator, hysteresis, self-oscillations, astrophysical research

A piezo actuator is used in astrophysics for image stabilization and scan system.^1–19 Frequency method for determination self-oscillations in scan system is applied.^20–46 for Nyquist stability criterion of self-oscillations at harmonious linearization of hysteresis characteristic of a piezo actuator.

Condition of self-oscillations

The scan system with a piezo actuator is used for astrophysical research in system adaptive optics. Nyquist stability criterion of self-oscillations at harmonious linearization of hysteresis characteristic^2,20–40 of a piezo actuator has the form

$W_l\left(\alpha \Omega \right)W_g\left(E_{m\text{max}}\right)=-1$

where α is the imaginary unit, Ω - the frequency of self-oscillations, $W_l\left(\alpha \Omega \right)$  - the frequency transfer function of the linear part, $W_g\left(E_{m\text{max}}\right)$  - the transfer function of the hysteresis part, $E_{m\text{max}}$  - amplitude of the electric field strength for m axis.

For the scan system with a piezo actuator for astrophysical research the condition of self-oscillations is written

$1+W_l\left(\alpha \Omega \right)W_g\left(E_{m\text{max}}\right)=0$

The condition of self-oscillations is determined in the form

$W_l\left(\alpha \Omega \right)=-\frac{1}{W_g\left(E_{m\text{max}}\right)}$

here the left side of this equation has the form of the amplitude-phase characteristic of the linear part of the system, and the right side of the equation has the form of the inverse amplitude-phase characteristic of the hysteresis link of the piezo actuator with the inverse sign  minus.

Preisach hysteresis function a piezo actuator has the form^22-40

$S_i=F{\left[{E_m|}_0^t,t,S_i\left(0\right),\text{sign}{\dot{E}}_m\right]}_{}^{}$

here $t,S_i,S_i\left(0\right),E_m$  and $\text{sign}{\dot{E}}_m$  - the time, the deformation, the initial deformation,, the strength of electric field and the sign velocity.

The symmetric hysteresis the deformation^22-40 a piezo actuator has the form

$S_i=d_{mi}{E}_m-{\gamma }_{mi}{E}_{m\text{max}}{\left(1-\frac{E_m^2}{E_{m\text{max}}^2}\right)}^n\text{sign}{\dot{E}}_m$

$d_{mi}=d_{mi}^0+a_{mi}{E}_m^2,{\gamma }_{mi}=S_i^0/E_{m\text{max}}$

here $d_{mi},{\gamma }_{mi},S_i^0,n$ , - the piezo module, the hysteresis coefficient, the relative deformation for $E_m=0$ , and the power 1, 2, 3, ….

The transfer function of the linear part of the scan system with a piezo actuator for elastic-inertia load^22,37-46 has the form

$W_l\left(p\right)=\frac{k_l}{T_t^2{p}^2+2T_t{\xi }_{t}p+1}$

After transformations we have this condition for the scan system with the PZT actuator at the power  in the form

$\frac{1}{\frac{1-T_t^2{\Omega }^2}{k_l}+\alpha \frac{2T_t{\xi }_t\Omega }{k_l}}=\frac{1}{-\left(d_{mi}^0+a_{mi}{E}_{m\text{max}}^2\right)+\alpha \frac{8{\gamma }_{mi}}{3\pi }}$

$\Omega =\frac{4{\gamma }_{mi}{k}_l}{3\pi T_t{\xi }_t}$

For the the scan system with the PZT actuator  = 3.2×10⁸ V/m, $d_{33}^0$  = 4×10^-10 m/V, ${\gamma }_{33}$  = 0.8×10^-10 m/V, $a_{33}$  = 3.1×10^-22 m³/V³, $T_t$  = 10^-3 s, ${\xi }_t$  = 10^-2 the frequency is determined $\Omega$  =1.1×10³ s^-1 with error of 10 %.

The frequency transfer function of the symmetric hysteresis the deformation of a piezo actuator is received in the form

$W_g\left(E_{m\text{max}}\right)=S_i\left(E_{m\text{max}}\right)/E_m\left(E_{m\text{max}}\right)$

$W_g\left(E_{m\text{max}}\right)=q_{mi}\left(E_{m\text{max}}\right)+\alpha {q^{\prime }}_{mi}\left(E_{m\text{max}}\right)$

For n = 1

$q_{mi}\left(E_{m\text{max}}\right)=d_{mi},{q^{\prime }}_{mi}\left(E_{m\text{max}}\right)=-\frac{4\cdot 2\cdot {\gamma }_{mi}}{\pi \cdot 3}=-\frac{8{\gamma }_{mi}}{3\pi }$

For n = 2

$q_{mi}\left(E_{m\text{max}}\right)=d_{mi},{q^{\prime }}_{mi}\left(E_{m\text{max}}\right)=-\frac{4\cdot 2\cdot 4\cdot {\gamma }_{mi}}{\pi \cdot 3\cdot 5}=-\frac{32{\gamma }_{mi}}{15\pi }$

For n = 2

$q_{mi}\left(E_{m\text{max}}\right)=d_{mi},{q^{\prime }}_{mi}\left(E_{m\text{max}}\right)=-\frac{4\cdot 2\cdot 4\cdot 6\cdot {\gamma }_{mi}}{\pi \cdot 3\cdot 5\cdot 7}=-\frac{192{\gamma }_{mi}}{105\pi }$

For n to n + 1

$q_{mi}\left(E_{m\text{max}}\right)=d_{mi},{q^{\prime }}_{mi\text{(}n\text{)}}\left(E_{m\text{max}}\right)=\frac{2n}{2n+1}{q^{\prime }}_{mi\text{(}n-1\text{)}}\left(E_{m\text{max}}\right)$

For n + 1

$q_{mi}\left(E_{m\text{max}}\right)=d_{mi},{q^{\prime }}_{mi}\left(E_{m\text{max}}\right)=-\frac{4\cdot 2\cdot 4\cdot 6\cdot \mathrm{...}\cdot 2n\cdot {\gamma }_{mi}}{\pi \cdot 3\cdot 5\cdot 7\cdot \mathrm{...}\cdot \left(2n+1\right)}$

The stability criterion and frequency method are used.

By using of frequency method the parameters of self-oscillations are obtained in the scan system. Nyquist stability criterion is used for calculation the self-oscillations in the control system with a piezo actuator at harmonious linearization of hysteresis characteristic of a piezo actuator.

For the scan system its condition of self-oscillations is determined. For calculation the self-oscillations frequency method is applied at harmonious linearization of hysteresis characteristic of a piezo actuator.

None.

The authors declare that there is no conflict of interest.

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